Unformatted text preview: probability of S on the n th trial is p n , we can determine whether or not the waiting time variable “ X = number of trials until the Frst S” is proper or not by applying the Sum-Product Lemma below. (The Sum-Product Lemma is a mathematical result from real analysis.) The Sum-Product Lemma: Suppose 0 < p n < 1 for each n . Then ∞ p n =1 (1-p n ) = c > if and only if ∞ s n =1 p n < ∞ application: in the special case (*) noted above, we have P ( X > n ) = P ( FF . . . F ) = (1-p 1 )(1-p 2 ) ··· (1-p n ) . Therefore f ( ∞ ) = lim n →∞ (1-p 1 )(1-p 2 ) ··· (1-p n ) = P ∞ n =1 (1-p n ), and so the Sum-Product Lemma can be applied to determine whether f ( ∞ ) > 0 or f ( ∞ ) = 0....
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- Winter '08
- Probability, Probability theory, Stochastic process, Proper random variables, C.D. Cutler